Optimal Distillation of Coherent States with Phase-Insensitive Operations

TL;DR

The paper tackles the problem of distilling pure coherent states from noisy coherent thermal inputs using phase-insensitive operations. It derives a universal bound on the asymptotic distillation error in terms of quantum information metrics and constructs an optimal non-Gaussian protocol that saturates this bound, revealing an operational interpretation of the purity of coherence via the Right-Logarithmic-Derivative Fisher information. A divide-and-distill strategy, combining concentration, a phase-insensitive non-Gaussian channel, and dilution, achieves the bound in the large-copy limit, while a Gaussian beam-splitter protocol remains suboptimal but practically implementable. The results elucidate fundamental limits of coherence distillation, connect them to a resource-theoretic viewpoint on asymmetry, and offer actionable insights for implementing phase-insensitive purification in optical settings.

Abstract

By combining multiple copies of noisy coherent states of light (or other bosonic systems), it is possible to obtain a single mode in a state with lesser noise, a process known as distillation or purification of coherent states. We investigate the distillation of coherent states from coherent thermal states under general phase-insensitive operations, and find a distillation protocol that is optimal in the asymptotic regime, i.e., when the number of input copies is much greater than 1. Remarkably, we find that in this regime, the error -- as quantified by infidelity (one minus the fidelity) of the output state with the desired coherent state -- is proportional to the inverse of the purity of coherence of the input state, a quantity obtained from the Right-Logarithmic-Derivative (RLD) Fisher information metric, hence revealing an operational interpretation of this quantity. The heart of this protocol is a phase-insensitive channel that optimally converts an input coherent thermal state with high amplitude, into an output with significantly lower amplitude and temperature. Under this channel, the purity of coherence remains asymptotically conserved. While both the input and desired output are Gaussian states, we find that the optimal protocol cannot be a Gaussian channel. Among Gaussian phase-insensitive channels, the optimal distillation protocol is a simple linear optical scheme that can be implemented with beam splitters.

Figures (10)

• Figure 1: Phase-insensitive distillation of coherent states -- The goal is to convert $n\gg1$ copies of a coherent thermal state$\rho(\beta,\alpha)=D(\alpha)\rho_{\text{th}}(\beta)D(\alpha)^\dag$, where $D(\alpha)$ is the Weyl displacement operator, to a pure coherent state $|\alpha\rangle$, or more generally $|s\alpha\rangle$ for some constant $s\ll n$, using phase-insensitive operations. The state $\rho(\beta,\alpha)$ describes, e.g., the output of a thermal attenuator channel given a pure coherent state $|\alpha_0\rangle$ as the input. The phase-insensitivity of the distillation operation ensures that the final output state is in-phase, i.e., synchronized with the input states $\rho(\beta,\alpha)$. We then ask, what is the optimal distillation protocol that achieves the highest fidelity with the desired state $|s \alpha\rangle$, and what is the value of this optimal fidelity?
• Figure 2: Performance of different distillation protocols -- This plot shows the infidelity factor$\delta^{\mathcal{E}}(\beta, \alpha)$, defined in Eq.(\ref{['eqn: intro delta']}), for different distillation protocols, as a function of the mean number of thermal excitations $n_{\text{th}}(\beta)$ in the input coherent thermal states. $\delta^{\mathcal{E}}(\beta, \alpha)$ characterizes the asymptotic performance of a distillation protocol in the many-copy regime. Using the properties of the purity of coherence marvian2020coherence, one can show that $\delta^{\mathcal{E}}(\beta, \alpha)$ is lower bounded by ${n_\text{th}(\beta)}/{2}+{n_\text{th}(\beta)}/{[2+4n_\text{th}(\beta)]}$. We prove that this can be achieved with a novel distillation protocol introduced in this paper. The red and blue curves correspond respectively to a simple protocol discussed in Fig.\ref{['first real']}, that is realizable with linear optical elements, and to a measure-and-prepare protocol. See below for further details.
• Figure 3: Coherence distillation in phase space --$n$ copies of the coherent thermal state $\rho(\beta,\alpha)$ can be reversibly transformed to state $\rho(\beta,\sqrt{n}\alpha)$ via phase-insensitive channels. Hence, any coherence distillation protocol can be understood as a sequence of phase-insensitive channels $\mathcal{E}_n$ that transform the state $\rho(\beta, \sqrt{n}\alpha)$ to a state close to $\rho(\infty,\alpha)= |{\alpha}\rangle\!\langle\alpha|$. Both the input and the desired output states are described by Gaussian Wigner distributions, with $x$ and $p$ variances $\braket{\Delta x^2}_\text{in} = \braket{\Delta p^2}_\text{in} =n_\text{th}(\beta)+1/2$, and $\braket{\Delta x^2}_\text{out} = \braket{\Delta p^2}_\text{out} =1/2$, respectively. The centers of the two Gaussian distributions have radii $r_\text{in}=\sqrt{n}|\alpha|$ and $r_\text{out}=|\alpha|$, respectively. From a classical perspective, one may expect that under the rescaling $x\rightarrow x/\sqrt{n}$ and $p\rightarrow p/\sqrt{n}$, which is a phase-insensitive map, the input distribution can be transformed to the output, provided that $n$ is equal to or larger than the ratio of the variances, namely $2 n_\text{th}(\beta)+1$. That is, using $n\ge 2 n_\text{th}(\beta)+1$ copies of the input $\rho(\beta,\alpha)$, one should be able to obtain an exact copy of the coherent state $|\alpha\rangle$. Indeed, this is exactly the same bound one obtains by considering the ratio of QFI $F_H$ for the output and input states (see Sec. \ref{['section: universal bounds proof']}). However, the above rescaling cannot be implemented as a physical process. Indeed, unless $\alpha=0$ or $\beta=\infty$, one needs an infinite number of copies of $\rho(\beta,\alpha)$ to obtain an exact copy of the pure coherent state $|\alpha\rangle$. As shown in marvian2020coherence, this can be established using the properties of the purity of coherence $P_H$. Furthermore, as we show in this letter, in the regime $n \gg 1$, this quantity determines the minimum achievable error in the output state (see Eq.(\ref{['opt bound']})). It is also worth noting that the above rescaling can be realized with a beam splitter, with an order $\mathcal{O}(n^{-1})$ correction that comes from the vacuum noise in the other input mode (See Fig. \ref{['first real']}).
• Figure 4: A suboptimal coherence distillation protocol using beam splitters -- Two different realizations of a distillation protocol that consumes $n$ copies of coherent thermal state $\rho(\beta, \alpha)$ to produce single mode $\rho(\beta_\text{out}, \alpha)$ in a lower temperature are presented (the change in temperature is indicated using darker and lighter shades of red for hot and cold states, respectively). This protocol achieves the optimal performance among Gaussian distillation protocols (see Proposition \ref{['prop: gaussian']}). The transmission/reflection coefficients of each of the beam splitters are adjusted to attain the state transformations indicated in the figure as per Eq.(\ref{['eq: bs alpha']}). First Realization -- In the first stage of this realization the coherence in $n$ input modes is concentrated into a single mode $\rho(\beta,\alpha)^{\otimes n} \longrightarrow \rho(\beta,\sqrt{n}\alpha)$ (we refer to this passive transformation as the concentration map; see Sec. \ref{['subsec: concentration dilution reversibility']} for further discussion). Then, to reduce the temperature we combine this state with an ancilla mode in a lower temperature $\beta_\text{cold}$ to realize $\rho(\beta,\sqrt{n}\alpha)\otimes \rho(\beta_\text{cold},0)\ \longrightarrow \ \rho(\beta',\alpha)\ ,$ where $n_\text{th}(\beta')$ is given by Eq.(\ref{['eq: bs n']}). If the ancilla mode is in the vacuum state, (i.e., $n_\text{th}(\beta_{\text{cold}})=0$), then $n_\text{th}(\beta')=n_\text{th}(\beta)/n$. From Eq.(\ref{['eqn: infid formula']}), for $n\gg 1$, the infidelity with the desired state $|\alpha\rangle$ is $\epsilon_n={n_\text{th}(\beta)}/{n}+\mathcal{O}(n^{-2})$. Second Realization -- In this realization, a cold thermal ancilla mode in the initial state $\rho_\text{th}(\beta_{\text{cold}})$ sequentially and weakly interacts with each copy of the state $\rho(\beta,\alpha)$ where $\beta_\text{cold}> \beta$ (See Appendix \ref{['appendix: gaussian prop']} for further discussion). Each interaction displaces the state of ancilla by $\approx \alpha/\sqrt{n}$.
• Figure 5: Optimal Distillation using the Divide and Distill Strategy -- The circuit above illustrates the optimal protocol, along with the 'divide and distill' strategy highlighted within the dashed box: Suppose the phase-insensitive channel $\mathcal{K}_n$ distills an approximate pure state $\sigma_n \approx |{\gamma_\text{out}}\rangle\!\langle\gamma_\text{out}|$ from $\rho(\beta, \gamma_\text{in})$ in the regime where $|\gamma_\text{out}|\ll1$ and $|\gamma_\text{in}|\gg1$ (discussed in Sec. \ref{['subsection: weak output regime']}). We can extend this optimal behavior to general inputs, i.e., $\rho(\beta, \sqrt{n} \alpha)$ for any $\alpha$, by first diluting the input into an appropriate number of copies ($B_n$) such that each copy is in the strong-input weak-output regime for $n \gg 1$, then applying $\mathcal{K}_n$ in parallel to each of the $B_n$ copies, and finally concentrating all these outputs $\sigma_n$ into a single mode $\sigma'_n$. In the limit of large $n$, this procedure achieves the highest possible fidelity $\braket{\alpha|\sigma'_n|\alpha}$ (discussed in Sec. \ref{['subsection: optimal general']}). It is worth noting that the first step of concentrating $\rho(\beta, \alpha)^{\otimes n} \leftrightarrow \rho(\beta, \sqrt{n}\alpha)$ is a reversible transformation (discussed in Sec. \ref{['subsec: concentration dilution reversibility']}).

Theorems & Definitions (18)

• Proposition 1
• Lemma 2
• Lemma 3
• Theorem 4
• Corollary 5
• proof
• Lemma 6
• Theorem 7
• Lemma 8
• Lemma 9